Asymptotic Analysis - Volume 88, issue 3

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ISSN 0921-7134 (P)
ISSN 1875-8576 (E)

Impact Factor 2018: 0.748

The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.

Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.

Abstract: Let X be an arbitrary Banach space. This work deals with the asymptotic behavior, the continuity and the compactness properties of solutions of the non-linear Volterra difference equation in X described by u(n+1)=λΣj=−∞ n a(n−j)u(j)+f(n,u(n)), n∈Z, for λ in a distinguished subset of the complex plane, where a(n) is a complex summable sequence and the perturbation f is a non-Lipschitz nonlinearity. Concrete applications to control systems and integro-difference equations are given.

Abstract: Let Ω⊂RN (N≥2) be a bounded C2 domain containing 0, 0<α<1 and 0<p<N/(N−2α). If δ0 is the Dirac mass at 0 and k>0, we prove that the weakly singular solution uk of (Ek ) (−Δ)α u+up =kδ0 in Ω, which vanishes in Ωc , is a classical solution of (E* ) (−Δ)α u+up =0 in Ω\{0} with the same outer data. Let A=[N/2α,1+2α/N) for N=2,3 and (√5−1)/4N<α<1, otherwise, A=∅; we derive that uk converges to ∞ in whole Ω as k→∞ for p∈(0,1+2α/N)\A, while the limit of uk is a strongly singular solution…of (E* ) for 1+2α/N<p<N/(N−2α).
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